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Technical Briefs

# Closed-Loop Dynamic Analysis of a Rotary Inverted Pendulum for Control Design

[+] Author and Article Information
Hashem Ashrafiuon1

Director, Center for Nonlinear Dynamics and Control Professor e-mail: hashem.ashrafiuon@villanova.edu Department of Mechanical Engineering, Villanova University, Villanova, PA 19085

Alan M. Whitman

Director, Center for Nonlinear Dynamics and Control Professor e-mail: hashem.ashrafiuon@villanova.edu Department of Mechanical Engineering, Villanova University, Villanova, PA 19085

1

Corresponding author.

J. Dyn. Sys., Meas., Control 134(2), 024503 (Dec 30, 2011) (9 pages) doi:10.1115/1.4005358 History: Received February 02, 2011; Revised September 26, 2011; Published December 30, 2011; Online December 30, 2011

## Abstract

This paper presents an approximate analytical solution for the weakly nonlinear closed-loop dynamics of the sliding phase of a sliding mode controlled rotary inverted pendulum based on the multiple scale method. A locally stable nonlinear sliding mode control law with starting configurations above the horizontal line is presented for the rotary inverted pendulum. The analytical expressions derived from the nonlinear solution of the reduced-order closed-loop dynamics provide both qualitative and quantitative insight into the closed-loop response leading to proper selection of parameters that guarantee stabilization and improve controller performance. The approximate analytical solution is verified through comparison with the exact numerical solution. The control performance predicted by the analytical solution is experimentally demo.

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## Figures

Figure 1

The rotary inverted pendulum

Figure 2

Comparison of simulation and experimental results; (top) pendulum rod angle, (middle) rotating arm angle, (bottom) control input voltage

Figure 3

Comparison of approximate and exact solutions for small δ  = 0.01; (top) u corresponding to φ , (bottom) x corresponding to θ

Figure 4

Comparison of approximate and exact solutions for large δ  = 0.1; (top) u corresponding to φ , (bottom) x corresponding to θ

Figure 5

Sensitivity of the closed-loop response to ɛ ; (top) u corresponding to φ , (bottom) x corresponding to θ

Figure 6

Sensitivity of the closed-loop response to Δ; (top) u corresponding to φ , (bottom) x corresponding to θ

Figure 7

Experimental control with varying ɛ values; (top) u corresponding to φ , (middle) x corresponding toθ , (bottom) input voltage v

Figure 8

Experimental control with varying Δ values; (top) u corresponding to φ , (middle) x corresponding to θ , (bottom) input voltage v

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